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994:. For example, the Trilinear polar of a point on the circumcircle must pass through its perspector, the Symmedian point X(6).
39:
of the triangle. "Although it is called a polarity, it is not really a polarity at all, for poles of concurrent lines are not
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Animation illustrating the fact that the locus of the trilinear poles of a pencil of lines passing through a fixed point
47:(1788–1867), a French engineer and mathematician, who introduced the idea of the trilinear polar of a point in 1865.
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not lying on the sides of the triangle and lines in the plane of the triangle not passing through the
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be any point in the plane of the triangle not lying on the sides of the triangle. Briefly, the
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Trilinear polars are not defined for points coinciding with the vertices of triangle
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953:. Thus the locus of the poles of a pencil of lines passing through a fixed point
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Axis of perspectivity of a given triangle, its cevian triangle, and some point
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932:{\displaystyle {\frac {x_{0}}{x}}+{\frac {y_{0}}{y}}+{\frac {z_{0}}{z}}=0.}
842:{\displaystyle {\frac {x_{0}}{X}}+{\frac {y_{0}}{Y}}+{\frac {z_{0}}{Z}}=0.}
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232:. The line of collinearity is the axis of perspectivity of triangle
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are perspective. The polar triangle is bounded by the tangents to
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752:{\displaystyle {\frac {x}{X}}+{\frac {y}{Y}}+{\frac {z}{Z}}=0.}
407:{\displaystyle {\frac {x}{p}}+{\frac {y}{q}}+{\frac {z}{r}}=0.}
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used this idea to define the concept of trilinear polars.
348:. Then the trilinear equation of the trilinear polar of
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is a certain correspondence between the points in the
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be the pole of a line passing through a fixed point
261:can also be obtained as the harmonic conjugates of
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573:Some of the trilinear polars are well known.
488:lines, which intersect at the trilinear pole
330:Let the trilinear coordinates of the point
60:Construction of a trilinear polar of a point
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648:is a circumconic of the reference triangle.
427:Construction of a trilinear pole of a line
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979:and the polar triangle with respect to
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120: Constructed trilinear polar (line
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315:with respect to the reference triangle
296:with respect to the reference triangle
515:respectively. Let the pairs of lines
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292:is the trilinear polar of the point
265:with respect to the pairs of points
250:is the trilinear polar of the point
1149:. MathWorld—A Wolfram Web Resource
1126:. MathWorld—A Wolfram Web Resource
1103:. MathWorld—A Wolfram Web Resource
1080:. MathWorld—A Wolfram Web Resource
561:is the trilinear pole of the line
436: Given trilinear polar (line
418:Construction of the trilinear pole
14:
1022:. Springer. pp. 102–103.
961:of the triangle of reference.
1:
946:of the triangle of reference
1179:Isotomic conjugate of a line
136:be a plane triangle and let
697:. Equation of the line is
674:with trilinear coordinates
656:with trilinear coordinates
613:The trilinear polar of the
595:The trilinear polar of the
577:The trilinear polar of the
549:are in perspective and let
193:with reference to triangle
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762:Since this passes through
189:is the cevian triangle of
1020:The Real Projective Plane
636:Poles of pencils of lines
111:lines which intersect at
1177:Geometrikon page :
1171:Geometrikon page :
1043:Coxeter, H.S.M. (2003).
1018:Coxeter, H.S.M. (1993).
200:. Let the pairs of line
170:In detail, let the line
555:center of perspectivity
182:respectively. Triangle
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1051:. Springer. pp.
968:is the perspector of
964:It can be shown that
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569:Some trilinear polars
446: Given triangle
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150:axis of perspectivity
69: Given triangle
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174:meet the sidelines
45:Jean-Victor Poncelet
1145:Weisstein, Eric W.
1122:Weisstein, Eric W.
1099:Weisstein, Eric W.
1076:Weisstein, Eric W.
1047:Projective Geometry
987:at the vertices of
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852:Thus the locus of
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326:Trilinear equation
222:Desargues' theorem
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25:trilinear polarity
21:Euclidean geometry
1194:Triangle geometry
1078:"Trilinear Polar"
957:is a circumconic
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1173:Trilinear polars
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1147:"Polar Triangle"
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1166:External links
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496:Let the line
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239:and triangle
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1151:. Retrieved
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1128:. Retrieved
1124:"Perspector"
1117:
1105:. Retrieved
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1082:. Retrieved
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603:of triangle
601:Lemoine axis
584:
581:of triangle
572:
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535:. Triangles
526:
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504:of triangle
495:
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344:
340:
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311:of the line
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288:If the line
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246:. The line
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944:circumconic
619:orthic axis
615:orthocenter
257:The points
51:Definitions
1153:3 February
1130:3 February
998:References
942:This is a
502:BC, CA, AB
176:BC, CA, AB
172:AP, BP, CP
43:." It was
230:collinear
1188:Category
1107:8 August
689: :
682: :
665: :
661: :
579:centroid
531:meet at
343: :
339: :
283:Poncelet
37:vertices
33:triangle
1084:31 July
617:is the
599:is the
588:is the
553:be the
533:U, V, W
513:X, Y, Z
263:D, E, F
259:X, Y, Z
226:X, Y, Z
218:X, Y, Z
180:D, E, F
152:of the
148:is the
1059:
1026:
527:AX, BY
523:CZ, AX
519:BY, CZ
486:Cevian
484:
482:
458:
456:
444:
434:
212:DE, AB
208:CA, FD
204:BC, EF
118:
109:Cevian
107:
105:
81:
79:
67:
475:from
303:then
98:from
31:of a
29:plane
1155:2023
1132:2023
1109:2012
1086:2012
1057:ISBN
1024:ISBN
856:is
652:Let
542:and
525:), (
521:), (
352:is
277:A, B
275:), (
273:C, A
271:), (
269:B, C
228:are
210:), (
206:), (
129:Let
991:ABC
976:ABC
950:ABC
628:ABC
607:ABC
585:ABC
546:UVW
539:ABC
511:at
508:ABC
477:XYZ
472:ABC
468:of
465:UVW
450:ABC
438:XYZ
429:XYZ
334:be
319:ABC
300:ABC
248:XYZ
243:DEF
236:ABC
197:ABC
186:DEF
178:at
164:ABC
156:of
144:of
133:ABC
122:XYZ
95:ABC
91:of
88:DEF
73:ABC
19:In
1190::
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989:△
985:E
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970:E
966:K
959:E
955:K
948:△
924:=
919:z
914:0
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899:y
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890:y
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879:x
874:0
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854:P
834:=
829:Z
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820:z
814:+
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764:K
744:=
739:Z
736:z
731:+
726:Y
723:y
718:+
713:X
710:x
694:0
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631:.
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399:=
394:r
391:z
386:+
381:q
378:y
373:+
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365:x
350:P
345:r
341:q
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332:P
317:△
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